| L(s) = 1 | − 2.13i·3-s + (−1.94 + 1.10i)5-s + (2.35 + 1.19i)7-s − 1.56·9-s + 2.33i·11-s − 1.09·13-s + (2.35 + 4.15i)15-s − 4.98·17-s + 2.57·19-s + (2.56 − 5.03i)21-s − 6.04·23-s + (2.56 − 4.29i)25-s − 3.07i·27-s − 0.561·29-s − 6.59·31-s + ⋯ |
| L(s) = 1 | − 1.23i·3-s + (−0.869 + 0.493i)5-s + (0.891 + 0.453i)7-s − 0.520·9-s + 0.703i·11-s − 0.302·13-s + (0.608 + 1.07i)15-s − 1.20·17-s + 0.590·19-s + (0.558 − 1.09i)21-s − 1.25·23-s + (0.512 − 0.858i)25-s − 0.591i·27-s − 0.104·29-s − 1.18·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0459i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0459i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5052501233\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5052501233\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.94 - 1.10i)T \) |
| 7 | \( 1 + (-2.35 - 1.19i)T \) |
| good | 3 | \( 1 + 2.13iT - 3T^{2} \) |
| 11 | \( 1 - 2.33iT - 11T^{2} \) |
| 13 | \( 1 + 1.09T + 13T^{2} \) |
| 17 | \( 1 + 4.98T + 17T^{2} \) |
| 19 | \( 1 - 2.57T + 19T^{2} \) |
| 23 | \( 1 + 6.04T + 23T^{2} \) |
| 29 | \( 1 + 0.561T + 29T^{2} \) |
| 31 | \( 1 + 6.59T + 31T^{2} \) |
| 37 | \( 1 + 5.49iT - 37T^{2} \) |
| 41 | \( 1 + 8.48iT - 41T^{2} \) |
| 43 | \( 1 + 1.32T + 43T^{2} \) |
| 47 | \( 1 + 9.74iT - 47T^{2} \) |
| 53 | \( 1 + 8.58iT - 53T^{2} \) |
| 59 | \( 1 + 14.3T + 59T^{2} \) |
| 61 | \( 1 - 0.620iT - 61T^{2} \) |
| 67 | \( 1 - 4.71T + 67T^{2} \) |
| 71 | \( 1 + 11.9iT - 71T^{2} \) |
| 73 | \( 1 + 9.96T + 73T^{2} \) |
| 79 | \( 1 - 10.6iT - 79T^{2} \) |
| 83 | \( 1 - 3.86iT - 83T^{2} \) |
| 89 | \( 1 - 2.82iT - 89T^{2} \) |
| 97 | \( 1 - 14.9T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.405566915234760966050167142960, −7.68708202781635110552424967310, −7.26041399171616217898106762299, −6.59288135172430211306740766617, −5.58419456505152941343020456303, −4.59849858775270747702385228498, −3.73514185577513972307794471335, −2.31624434962970029193247962262, −1.83238696582190658251167410257, −0.17191932077718476955299550339,
1.44044747708907302721744601039, 3.04880894458018226638543327018, 3.99290528731802857526469622235, 4.50646971307650880770296224695, 5.09829284915587407191527744474, 6.13981928889710210904496845081, 7.37314165485462849742432194336, 7.938401810117933966101784747644, 8.749096037981035108413542086228, 9.320255571543442037331032864046
